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MSP101

MSP101 is an ongoing series of informal talks by visiting academics or members of the MSP group. The talks are usually Friday 4.15pm in room LT1310 in Livingstone Tower. They are announced on the msp-interest mailing-list. The list of talks is also available as a RSS feed and as a calendar file.

List of previous talks

The last few years has seen the development — largely in Canada and Australia — of an axiomatic approach to differential geometry based on tangent categories. Tangent categories incorporate the previous leading settings for differential geometry: finite dimensional manifolds, synthetic differential geometry, convenient manifolds, etc. In addition they widen the scope significantly as they also include combinatorial species, Goodwillie Functor calculi, and examples from computer science. The talk will give a survey of tangent categories and some of the developments so far.

2018-12-07, 16:00: Interacting Frobenius Algebras (Joe Collins, MSP)

What is a Frobenius algebra? What is a Hopf algebra? And why are they such good friends? In this talk, I will be talking about PROPs, what an interacting Frobenius algebra is, and some weird stuff that appears with them, and I shall be drawing lots of pretty pictures as well.

2018-11-30: Dual adjunctions (Simone Barlocco, MSP)

Coalgebras provide an abstract framework to represent state-based transition systems. Modal logics provide a formal language to specify such systems. In our recently submitted work (joint work with Clemens Kupke and Jurriaan Rot) we devise a general algorithm to learn coalgebras. Modal formulas are used as tests to probe the behaviour of states.

In my introductory talk, I will discuss how to set up a general framework that connects coalgebras and their modal logics via a dual adjunction. Moreover, I will show a known result which guarantees that indistinguishable states wrt to modal formulas are behavioural equivalent, a key fact that entails that — whenever possible — our algorithm learns a minimal representation of a coalgebra.

State-based transition systems are often studied relative to a specified initial state. System behavior then only depends on those states that are "reachable" from the initial state. This has both consequences for the theory (e.g. by allowing to prove non-definability results in modal logic) and practice (by making seemingly large systems more manageable) of such systems.

Coalgebras provide a general model for transition systems. In this introductory talk I will discuss how to define the reachable part of a coalgebra via the notion of T-base for an endofunctor T from [1]. We will first discuss this notion and then provide a sufficient categorical condition for the existence of the T-base. We will then show how to characterise the reachable part of a coalgebra as least fixpoint of a monotone operator.

Implicit Computational Complexity is the study of programming languages or logical systems that capture complexity classes. Roughly, every program that can be written in the language is in some complexity class. Many of the languages that have been proposed for capturing useful classes like PTIME are not much fun to program in. However, the work of the late Martin Hofmann included work on languages like LFPL, which only allows polynomial time computation, but is also reasonably usable. I'll talk about LFPL, and how the proof of polynomial time bounds works.

I will present a type theory whose judgements are indexed by a preorder of "worlds", representing for example nodes in a distributed computation, or a security level. This means a term may only typecheck at particular worlds, and will be mobile upwards along the preorder (for instance every low security value is also a high security one). To enable talking about the world structure without compromising mobility, the terms can talk about "shifts", which describe relative worlds.

I then give a semantic model based on the usual presheaf model for STLC where worlds form the base category, and shifts are endofunctors on the worlds. This semantics will show our programs are indeed oblivious to data they cannot "see". Examples will be given to demonstrate this framework in some concrete cases, and to motivate future work.

In this talk, I will describe a hierarchy of program transformers in which the transformer at each level builds on top of the transformers at lower levels. The program transformer at the bottom of the hierarchy corresponds to positive supercompilation, and that at the next level corresponds to distillation. I will then try to characterise the improvements that can be made at each level, and show how the transformers can be used for program verification and theorem proving.

By way of giving CS106 students better tools to tackle the concept of memory in circuits, I implemented a small programming language called (for reasons which are unlikely to become clear) Syrup. Syrup is suspiciously like a dialect of Haskell, except that the blessed monad allows bits of state. Marking homework done in Syrup necessitates checking whether two circuits have the same externally observable behaviour, which makes it a matter of bisimulation. I'll talk a bit about how much fun it was implementing the decision procedure to find either a bisimulation or a countermodel (for purposes of decent feedback).

Near-term quantum computers have many limitations which make them difficult to use for stuff. I will outline some of the difficulties and handwave at some new ideas from compositional mathematics that might help us address these problems.

2018-10-05, 16:00: ICFP trip report (James Wood, MSP)

Last week, I attended the ICFP conference and various associated workshops in St Louis. In this trip report, I will talk about selected talks and the people I met there. If time allows, I may also cover the adventures of Ioan, one of our summer interns and current undergraduate.

I'll do a 101 today, filed under "stuff everybody should know" about first-order unification (the algorithm underlying Hindley-Milner type inference, Prolog, etc). But then I'll throw in the twist of considering syntax with binding. The way I cook it, this makes essential use of the structure of the category of thinnings.

The minimalist tradition in type systems makes for easy mathematics, but often leaves their mechanisms needlessly obscured.

I build a structure for Hindley-Milner checking problems in the tradition of Type Inference in Context. This structure is derived from typing rules in the style of my first talk and mirrors data structures used for elaboration problems in dependent type systems — offering a notation that can be used among designers and implementors of type systems and even in explaining their behaviour to users.

The minimalist tradition in type systems makes for easy mathematics, but often leaves their mechanisms needlessly obscured.

One possible remedy is to track the behaviour of information in a system — its creation, its destruction and how it flows between constraints and source locations. I illustrate this with the Simply-Typed Lambda Calculus.

It is a well-known fact (used e.g. in model checking) that, on finitely branching transition systems, finite trace equivalence coincides with infinite trace equivalence. I will show how to prove this coinductively, which is arguably nicer than the standard inductive proof.

The paper Type-and-Scope Safe Programs and Their Proofs abstracts the common type-and-scope safe structure from computations on lambda-terms that deliver, e.g., renaming, substitution, evaluation, CPS-transformation, and printing with a name supply. By exposing this structure, we can prove generic simulation and fusion lemmas relating operations built this way. In this talk I will present this approach but for simpler setting of Hutton's Razor. This reduces the mathematical structures involved from relative structures to the ordinary counterparts.

Though Multi-Dimensional Arrays (MDAs) seem conceptually straightforward, it's not easy to come up with a mathematical theory of arrays that can be used within optimising compilers. We'd like to treat arrays as functions from indices to values with some domain restrictions. It is desirable that these domain restrictions are specified in a compact form, and are equipped with closed algebraic operations like intersection, union, etc. We are going to consider a few typical models like hyperrectangulars, grids and polyhedrons.

When typing array operations, ideally we'd like to find a balance between tracking all the shapes of arrays and allowing for generic array operations. This proves to be tricky, for reasons we'll explain.

We will propose, tentatively, an analysis of MDAs in terms of container functors. The aim is to supply concepts helpful when thinking about MDAs, e.g. when designing notations for coding with arrays. Some intriguing gadgetry shows up.

This is the second half of Helle's talk on a (co)algebraic treatment of Markov Decision Processes. It focuses on a coinductive explanation of policy improvement using a new proof principle, based on Banach's Fixpoint Theorem, that we call contraction coinduction.

2018-05-23, 16:00: How Do We Model a Problem Like Mutable State? (Bob Atkey, MSP)

Before getting lost in the realms of higher dimensions we should see wether we can interpret set-level HoTT. We know how to deal with functional extensionality and a static universe of propositions (see Observational Type Theory) but what about a dynamic universe of propositions, i.e. one reflecting HProps that also validates propositional extensionality. I will discuss the problems modelling this and a possible solution using globular setoids.

The dynamic prop corresponds to a subobject classifier in a topos (in particular we get unique choice) while the static universe corresponds to a quasitopos I am told.

In this talk, we study Markov decision processes (MDPs) with the discounted sums criterion from the perspective of coalgebra and algebra. Probabilistic systems, similar to MDPs but without rewards, have been extensively studied, also coalgebraically, from the perspective of program semantics. Here, we focus on the role of MDPs as models in optimal planning, where the reward structure is central. Our main contributions are: (i) a coinductive explanation of policy improvement using a new proof principle, based on Banach's Fixpoint Theorem, that we call contraction coinduction, and (ii) showing that the long-term value function of a policy can be obtained via a generalized notion of corecursive algebra, which takes boundedness into account.

A proof interpretation translates proofs of one logical system into proofs of another (example: the double-negation interpretation of classical logic into constructive logic). This often reveals some information about the original system (e.g. classical logic is equiconsistent with constructive logic). Gödel's Dialectica interpretation (named after the journal it was published in) translates Heyting arithmetic (the constructive theory of the natural numbers, including induction) into System T (the quantifier-free theory of the simply typed lambda calculus with natural numbers) — quantifier complexity is traded for higher type complexity. Combining this translation with (a refined) double negation translation, one can extract System T programs from a proof of a forall-exists statement, even if this proof is using non-constructive priciples such as Markov's Principle, Excluded Middle, or the Quantifier-Free Axiom of Choice. I've always found the Dialectica translation mystifying, so I'll try to explain the intuition behind it.

Higher inductive types (HITs) in Homotopy Type Theory (HoTT) allow the definition of datatypes which have constructors for equalities over the defined type. HITs generalise quotient types and allow to define types which are not sets in the sense of HoTT (i.e. do not satisfy uniqueness of equality proofs) such as spheres, suspensions and the torus. However, there are also interesting uses of HITs to define sets, such as the Cauchy reals, the partiality monad, and the internal, total syntax of type theory. In each of these examples we define several types that depend on each other mutually, i.e. they are inductive-inductive definitions. We call those HITs quotient inductive-inductive types (QIITs).

Although there has been recent progress on the general theory of HITs, there isn't yet a theoretical foundation of the combination of equality constructors and induction-induction, despite having many interesting applications. In the present paper we present a first step towards a semantic definition of QIITs. In particular, we give an initial-algebra semantics and show that this is equivalent to the section induction principle, which justifies the intuitively expected elimination rules.

In preparation for Fred's talk about "Quotient Inductive-Inductive Types" next week I will introduce quotients and setoids in type theory and some of the issues surrounding them. The use of setoids is discouraged by many doctors and can lead to a contagious and incurable condition: relation preservation. Quotients on the other hand are dangerous if not correctly handled and can lead to unsightly things appearing where they shouldn't such as inhabitants of the excluded middle.

Let us revisit the definition of a category and define it in a way which has the advantage that we can generalize it to higher dimensions. Why am I interested in higher categories (or specifically (\infty,1)-categories)? I have a few problems in Homotopy Type Theory which I think can be solved using these beasts: the coherence problem for type theory in type theory (in the moment I cannot even define the standard model) and generalizing the Hungarian approach to Quotient Inductive Inductive Types (QIITs) to Higher Inductive Inductive Types (HIITs).

The sets foundations of nominal techniques are usually taken to be Fraenkel-Mostowski set theory (which is ZFA + a finite support property). I will argue that in many situations, a new foundation which I call Equivariant ZFA (EZFA) may be a better choice, because you can do everything in EZFA that you can do in FM and furthermore EZFA with Choice (EZFAC) is consistent whereas FM with Choice is not.

I will define EZFA and how it interacts with Choice.

I will prove that EZFA is equivalent to ZFA.

I will then prove that EZFA is not equivalent to ZFA.

I will explain why I think EZFA(C) may be useful, why my last three papers were actually written in EZFAC, and finally I will discuss the implications this may have for mathematical foundations in general.

Quantum mechanics is dope, so it makes sense that making a computer using the principles of quantum mechanics would also be pretty sick. However, the formalism that is used by physicists, called a Hilbert space, is not specialised for this purpose. In particular, it is
1) Difficult to prove properties about programs for quantum computers
2) Difficult to see what is these programs are actually doing

Thankfully, category theory is very cool! Using ZX calculus, we can talk about quantum computing in a much clearer manner. I will be introducing some fundamental quantum mechanics and ZX calculus, and then using ZX calculus I will talk about Shor's algorithm.

The protocols that describe the interactions between IP Cores on System-on-a-Chip architectures are well-documented. These protocols described not only the structural properties of the physical interfaces but also the behaviour of the emanating signals. However, there is a disconnect between the design of SoC architectures, their formal description, and the verification of their implementation in known hardware description languages.

Within the Border Patrol project we are investigating how to capture and reason about the structural and behavioural properties of SoC architectures using state-of-the-art advances in programming language research. Namely, we are investigating using dependent types and session types for the description of hardware communication.

In this talk I will discuss my work in designing a linked family of languages that captures and reasons about the topological structure of a System-on-a-Chip. These languages provide correct-by-construction guarantees over the interaction protocols themselves; the adherence of a component that connects using said protocols; and the validity of the specified connections. These guarantees are provided through abuse of indexed monads to reason about resource usage; and general (ab)use of dependent types as presented in Idris.

I will not cover all aspects of the languages but will concentrate my talk detailing the underlying theories that facilitate the correct-by-construction guarantees.

Many program properties are relational, comparing the behavior of a program (or even two different programs) on two different inputs. While researchers have developed various techniques for verifying such properties for standard, deterministic programs, relational properties for probabilistic programs have been more challenging. In this talk, I will survey recent developments targeting a range of probabilistic relational properties, with motivations from privacy, cryptography, machine learning. The key idea is to meld relational program logics with an idea from probability theory, called a probabilistic coupling. The logics allow a highly compositional and surprisingly general style of analysis, supporting clean proofs for a broad array of probabilistic relational properties.

Adjoint functors arise everywhere, but how do we find them? It is a fun exercise to prove that right adjoints preserve limits, and, dually, that left adjoints preserve colimits. An adjoint functor theorem is a statement that (under certain conditions) the converse holds: a functor which preserves limits is a right adjoint. I will discuss the General Adjoint Functor Theorem, and why Peter Johnstone considers it fundamentally useless.

The coinductive proof method can be enhanced by several techniques, often referred to as up-to-techniques. I will talk about the basic theory of coinduction-up-to, and a little about the more recent notion of companion. This companion is the largest valid up-to technique for a given predicate, and gives a nice way of working with coinduction up-to.

In this talk, I will present a simple yet powerful principle for coinductive reasoning, which we call "parameterized coinduction". More specifically, it is as simple as the Knaster-Tarski theorem without requiring any syntactic checking, yet as powerful as Coq's syntactic guarded coinduction supporting incremental reasoning. As an important consequence, parameterized coinduction can easily support complex nested induction-coinduction.

We also implemented the parameterized coinduction as the Coq library called "paco", which can be found at:

My talk will be based on our recent paper "A Compositional Treatment of Iterated Open Games". In this paper we introduce a new operator on open games to capture the concept of subgame perfect equilibrium as well as providing final coalgebra semantics of infinite games.

I'll talk about a way of measuring the sizes of trees using weighted tree automata, in a compositional way that works well with pattern matching. This is based on some work by Georg Moser and Martin Hofmann.

I talk about three methods for measuring the size of formulas in the modal mu-calculus and explore how the choice between them influences the complexity of computations on formulas. Especially, I focus on the guarded transformation, which is a simple syntactic transformation on formulas that is commonly assumed to be polynomial but has recently been shown to require exponential time.

I will complain about the mess in the literature and present two of our (Clemens, Yde and me) own preliminary results:

1) There is a polynomial guarded transformation if we measure the input formula in the number of its subformulas and measure the output formula in the size of its closure.

2) If there is a polynomial guarded transformation where we measure the input formula in the size of its closure then there is a polynomial algorithm for solving parity games. Hence finding such a transformation is at least as hard as solving parity games, which is commonly believed to be quite hard.

We employ an automata-theoretic approach that relates the different measures for the size of a formula to different constraints on the transition structure of an automaton corresponding to the formula.

This is a very technical talk but there will be many pictures!

2017-10-13, 16:00: Coalgebraic Learning (Simone Barlocco, MSP)

Automata learning is a well known technique to infer a finite state machine from a set of observations. One important algorithm for automata learning is the L* algorithm by Dana Angluin. In this 101 I will present a new perspective on L* using ideas from coalgebra and modal logic. After a brief recap of how L* works, I will describe a generalisation of the L* algorithm to the coalgebraic level. I will conclude my talk with two concrete instances of the general framework: the learning of Mealy machines and of bisimulation quotients of probabilistic transition systems. Joint work with Clemens Kupke.

Building on last week's introduction to Martin-Löf 1971, we describe a toy type theory which not only accounts for what type terms have, but also where they live. This extra information can be interpreted as where (physically) data lives, at what phase (typechecking vs runtime) it exists, when it exists, or who has access to the data.

In return for caring about these "worlds" describing where data lives, we get applications to distributed computing and erasure for efficient code generation, with future work to consider productivity and security.

Per Martin-Löf's 1971 Theory of Types is the ancestor of the type systems used today in Agda, Coq, Idris, NuPRL, and many other variations on the theme of dependent types. Its principal virtue is its simplicity: it has very few moving parts (but they move quite a lot). Its well known principal vice is its inconsistency: you can write a looping program inhabiting any type (thus 'proving' any proposition). I'll be talking about the design principles for constructing dependent type systems which are bidirectional — clearly split into a type checking part and a type synthesis part. By following these principles, it gets easier to establish good safety properties of these systems. In particular, I'll sketch how to keep type safety ("well typed programs don't go wrong") separate from normalization ("all computations terminate"). Martin-L&oumlf's 1971, reformulated bidirectionally, makes a good example, because it's small and type-safe, but not normalizing.

Cubical Type Theory (CTT) provides an extension of Martin-Löf Type Theory (MLTT) where we can interpret the univalence axiom while preserving the canonicity property, i.e. every closed term actually computes to a value. The typing and equality rules of CTT come as a fairly well-behaved extension of the ones of MLTT and the denotational model and prototype implementation help clarifying the system further.

Given the above it felt reasonable to introduce the features of CTT into a more mature proof assistant like Agda, and this talk reports the status of this endeavour. In short:

The univalence axiom is proven as a theorem and we successfully tested its computational behavior on small examples.

comp computes for any parametrized data or record types, including coinductive ones, but it is stuck for inductive families.

The interaction of the path type and copatterns gives extensionality principles for coinductive records.

The interval I is an actual type, we also have restriction types A[φ &mapsto; u] and types for partial elements Partial φ A. Their sort makes sure comp does not apply to them.

In the categorical semantics of (e.g.) the simply typed lambda calculus substitution of a variable by a term is achieved by composing morphisms. What is the equivalent notion in diagrammatic languages? What even is a "variable" in this context? I'll sketch some (pretty) rough ideas for the beginnings of a “functional language” of diagrams including substitution, binding, and pattern matching. It turns out to all be about operads and co-operads.

Social, biological and economic networks grow and decline with recurrent fragmentation and re-formation, often explained in terms of external perturbations. I will present a model of dynamical networks and evolutionary game theory that explains these phenomena as consequence of imitation and endogenous conflicts between "cooperators" and "cheaters". Cooperators promote well-connected prosperous (but fragile) networks and cheaters cause the network to fragment and lose its prosperity. Once the network is fragmented it can be reconstructed by a new invasion of cooperators, leading to recurrent cycles of formation and fragmentation observed, for instance, in bacterial communities and socio-economic networks. In the last part of the talk, I will briefly introduce my current works on the role of individual decision-making in cooperative communities and the possibility of synthetic biology to address these ideas in microbial communities.

Enumeration of graphs on surfaces (or "maps") is an active topic of research in combinatorics, with links to wide-ranging domains such as algebraic geometry, knot theory, and mathematical physics. In the last few years, it has also been found that map enumeration is related to the combinatorics of lambda calculus, with various well-known families of maps in 1-to-1 correspondence with various natural families of linear lambda terms. In the talk I will begin by giving a brief survey of these enumerative connections, then use those to motivate a closer look at the surprisingly rich topological and algebraic properties of linear lambda calculus.

At last year's WadlerFest celebration, Conor presented a dependent type theory where variables are tagged with information about how they are used. Variable usage tagging has been developed in the non dependent setting, starting with Girard's Linear Logic, and culminating with recent work in contextual effects, coeffects, and quantitative type theories. The subtlety with dependent types lies in how to account for the difference between usage in types and terms. Conor's system handles this by treating usage in types as a "zero" usage, so that it doesn't affect the usage in terms. This is a departure from previous "linear" type theories that maintains a strict separation between usage controlled information, which types cannot depend on, and unrestricted information, which types can depend on.

Conor presented a syntax and typing rules for the system, as well as an erasure property that exploits the difference between "not used" and "used", but doesn't do anything with the finer usage information available. I'll present a collection of models for the system that fully exploit the usage information. This will give interpretations of type theory in resource constrained computational models, Geometry of Interaction models, and imperative models. To maintain order, I will gather all these notions of model under a new concept of "Quantitative Category with Families", a generalisation of the standard "Category with Families" class of models of dependent types.

This is an informal talk on the interesting properties I've found when playing with the unknotting problem (knot simplification moves that help to establish whether any given knot is a loop in complicated disguise, or something really knotted).

I'll discuss the syntax that I've used for annotating knots that leads to a(n almost) syntax based method for unknotting, but that hints further at unknotting in a more interesting way by using an unintentional property of the syntax.

I'll present some examples of the problems with representing knots and how the syntax and reduction rules help, in my opinion, to make unknotting more tangible.

The Kameda-Weiner algorithm takes a machine (nondeterministic finite automaton) as input, and provides an optimal machine (state-minimal nondeterministic finite automaton) as output. In this talk I will discuss work which provides a clear explanation of it, by translating the various syntactic constructs into more meaningful order-theoretic ones, and then composing them together to prove correctness.

A key result in computational learning theory is Dana Angluin's L* algorithm that describes how to learn a regular language, or a deterministic finite automaton (DFA), using membership and equivalence queries. In my talk I will present a generalisation of this algorithm using ideas from coalgebra and modal logic — please note, however, that prior knowledge of these topics will not be required.

In the first part of my talk I will recall how the L* algorithm works and establish a link to the notion of a filtration from modal logic. Furthermore I will provide a brief introduction to coalgebraic modal logic. In the second part of my talk I will present a generalisation of Angluin's original algorithm from DFAs to coalgebras for an arbitrary finitary set functor T in the following sense: given a (possibly infinite) pointed T-coalgebra that we assume to be regular (i.e. having an equivalent finite representation) we can learn its finite representation by (i) asking "logical queries" (corresponding to membership queries) and (ii) making conjectures to which a teacher has to reply with a counterexample (equivalence queries). This covers (known variants of) the original L* algorithm and algorithms for learning Mealy and Moore machines. Other examples are infinite streams, trees and bisimulation quotients of various types of transition systems.

1. define an internal syntax of Type Theory without reference to untyped preterms;
2. define a version of the partiality monad that doesn't require countable choice.

On the one hand I think that these applications are interesting because they represent applications of HoTT which have nothing to do with homotopy theory; on the other hand they are clearly not very higher order (in the sense of truncation levels) but can be defined in the set-truncated fragment of HoTT. Hence my question: what are interesting applications of higher types which are not directly related to synthetic homotopy theory?

This talk is based on joint work with Paolo Capriotti, Nils Anders Danielsoon, Gabe Dijkstra, Ambrus Kaposi and Nicolai Kraus.

A traditional source of complaint from CS undergraduates (especially in the USA, but in other places, too) is that they are made to learn too much standard issue mathematics with little apparent relevance to computation. Differential calculus (with its usual presentational focus on physical systems) is often picked upon as the archetype. What we see in action is the fragile male ego: they are not so quick to complain about the unimportance of things they do not find difficult. All of which makes more delicious the irony that differential operators have a key role to play in understanding discrete structures, such as automata, datatypes, execution stacks, and plenty more.

The basic idea is as follows: to put your finger over any single K in the pair of words

BREKEKEKEX KOAXKOAX

you must choose either to put your finger over a single K in BREKEKEKEX and pair with KOAXKOAX intact, or to leave BREKEKEKEX intact and cover a K in KOAXKOAX. You have just followed Leibniz's rule for differentiating a product (with respect to K), and computed a one-hole context for a K in a data structure.

Newton, of course, would point out that such derivatives arise as the limit of a divided difference, a concept worthy of study in more generality. I would point out that divided differences are often definable, even in situtations when neither division nor difference makes much apparent sense. Notably, Brzozowski's derivative for regular languages is a divided difference (even though it is not Leibniz's derivative).

I'll work mainly with containers (which look a lot like power series) but make sure there are plenty of concrete examples. In practice, it becomes rather useful to compute derivatives by pattern matching on types, which is especially funny as symbolic differentiation is the first example in the literature of computing anything by pattern matching at all.

I'll report on my attempts to design a cubical type theory together with Dan Licata and Ed Morehouse during my visit to Wesleyan University, Middletown, Connecticut. We had something which seemed quite promising, but that falls apart just short of the finish line; I'll tell you about it in the hope of miraculous rescue from the audience. However, I'll start from basics so that everyone has a chance to join in in the fun. Mentions of Donald Trump will be kept to a minimum.

PCF is the prototypical functional programming language, with two data types (naturals and booleans), lambda-abstraction and recursion. PCF was introduced by Gordon Plotkin in his seminal "LCF Considered as a Programming Language" paper from 1977. Despite PCF's simplicity, its semantics is theoretically interesting. I will introduce PCF, its operational semantics, the "standard" domain-theoretic denotational semantics and show that the two agree on closed programs. Finally, I will discuss observational equivalence for PCF and show that the denotational semantics fails to be "fully abstract".

Hoare Logic is a logic for proving properties of programs of the form: if the initial state satisfies a precondition, then the final state satisfies a postcondition. Hoare logic proofs are structured around the structure of the program itself, making the system a compositional one for reasoning about pieces of programs. I'll introduce Hoare Logic for a little imperative language with WHILE loops. I'll then motivate Separation Logic, which enriches Hoare Logic with a Frame Rule for local reasoning.

Blockchains, i.e. decentralised, distributed data structures which can also carry executable code for a non-standard execution environment, introduce new models of computation. Decentralised, here, means, informally speaking, "without central control", e.g. a currency without a (central) bank, but much more. Blockchains support the recently introduced virtual currencies, a la Bitcoin, and a new class of decentralised applications, including smart contracts. In this talk we will introduce the main aspects of a blockchain, with particular reference to the Bitcoin blockchain as a paradigmatic case of such a new model of computation, and also touching on smart contracts. No previous knowledge of bitcoin/blockchain required for this introductory talk.

In this 101 I plan to discuss omega-automata, i.e., finite automata that operate on infinite words/streams. These automata form an important tool for the specification and verification of the ongoing, possibly infinite behaviour of a system. In my talk I will provide the standard definition(s) of omega-automata and highlight what makes omega-automata difficult from a coalgebraic perspective. Finally, I am going to discuss the work by Ciancia & Venema that provides a first coalgebraic representation of a particular type of omega-automata, so-called Muller automata.

Statistical models in e.g. machine learning are traditionally expressed in some sort of flow charts. Writing sophisticated models succinctly is much easier in a fully fledged programming language. The programmer can then rely on generic inference algorithms instead of having to craft one for each model. Several such higher-order functional probabilistic programming languages exist, but their semantics, and hence correctness, are not clear. The problem is that the standard semantics of probability theory, given by measurable spaces, does not support function types. I will describe how to get around this.

2017-02-09, 14:00: MSP 101: Automata learning (Simone Barlocco, MSP)

Automata learning is a well known technique to infer a finite state machine from a set of observations. One important algorithm for automata learning is the L* algorithm by Dana Angluin. In this 101, I will explain how the L* algorithm works via an example. Afterwards, I will discuss the ingredients of the algorithm both in the standard framework by Angluin and in a recently developed categorical/coalgebraic framework by Jacobs & Silva. Lastly, I plan to outline the proof of the minimality of the automaton that is built by the learning algorithm.

We compare the expressive power of three programming abstractions for user-defined computational effects: Bauer and Pretnar's effect handlers, Filinski's monadic reflection, and delimited control. This comparison allows a precise discussion about the relative merits of each programming abstraction.

We present three calculi, one per abstraction, extending Levy's call-by-push-value. These comprise syntax, operational semantics, a natural type-and-effect system, and, for handlers and reflection, a set-theoretic denotational semantics. We establish their basic meta-theoretic properties: adequacy, soundness, and strong normalisation. Using Felleisen's notion of a macro translation, we show that these abstractions can macro-express each other, and show which translations preserve typeability. We use the adequate finitary set-theoretic denotational semantics for the monadic calculus to show that effect handlers cannot be macro-expressed while preserving typeability either by monadic reflection or by delimited control. We supplement our development with a mechanised Abella formalisation.

In this 101 I outline the syntax and semantics of classical first order predicate logic. I try to also mention some of the characteristic properties of first order logic such as compactness, the Löwenheim-Skolem theorem or locality properties in finite model theory.

We explore the design and implementation of Frank, a strict functional programming language with a bidirectional effect type system designed from the ground up around a novel variant of Plotkin and Pretnar's effect handler abstraction.

Effect handlers provide an abstraction for modular effectful programming: a handler acts as an interpreter for a collection of commands whose interfaces are statically tracked by the type system. However, Frank eliminates the need for an additional effect handling construct by generalising the basic mechanism of functional abstraction itself. A function is simply the special case of a Frank operator that interprets no commands.

Moreover, Frank's operators can be multihandlers which simultaneously interpret commands from several sources at once, without disturbing the direct style of functional programming with values.

Effect typing in Frank employs a novel form of effect polymorphism which avoid mentioning effect variables in source code. This is achieved by propagating an ambient ability inwards, rather than accumulating unions of potential effects outwards.

I'll give a tour of Frank through a selection of concrete examples.

(Joint work with Conor McBride and Craig McLaughlin)

2016-12-14, 11:00: Compositional Game Theory (Alasdair Lambert, MSP)

I will be discussing composition in a model of economic game theory and methods for representing the impact of choice on subsequent games. Time permitting I will also work through some games using this model.

A filter P is a consistent deductively closed set of predicates. A filter is prime when

(φ ∨ ψ) ∈ P ⇒ (φ ∈ P ∨ ψ ∈ P)

In words: if phi-or-psi is in P then phi is in P or psi is in P. Primeness gives soundness for disjunction.

Using this it is not hard to construct a semantics to propositional logic in which a predicate φ "means" the set of prime filters containing it. This is a standard "trick" for building semantics and is an extremely useful proof-method.

I have developed a semantics for predicate logic and also for the lambda-calculus based on similar notions of filter, but in a nominal context — meaning that filters are developed using Fraenkel-Mostowski (FM) set theory instead of Zermelo-Fraenkel (ZF) set theory. What matters here is that FM sets have additional name structure over ZF sets, and this additional structure can be exploited to give semantics to the extra structure that predicates have over propositions, and in particular the additional name structure lets us write down primeness conditions for soundness for universal quantification.

The resulting semantics is rich and interesting. In a sentence: nominal techniques help us to extend the notion of Stone representation and duality from propositional logic to full first-order logic (also with equality, if we wish, and also to other logics and calculi with variables and quantifiers).

We discuss a number of semantic properties pertaining to formulas of the modal mu-calculus. For each of these properties we provide a corresponding syntactic fragment, in the sense that a mu-calculus formula \phi has the given property iff it is equivalent to a formula \phi' in the corresponding fragment. Since this formula \phi' will always be effectively obtainable from \phi, as a corollary, for each of the properties under discussion, we prove that it is decidable in elementary time whether a given mu-calculus formula has the property or not.

The properties that we study have in common that they all concern the dependency of the truth of the formula at stake, on a single proposition letter p. In each case the semantic condition on \phi will be that \phi, if true at a certain state in a certain model, will remain true if we restrict the set of states where p holds, to a special subset of the state space. Important examples include the properties of complete additivity and (Scott) continuity, where the special subsets are the singletons and the finite sets, respectively.

Our proofs for these characterisation results will be automata-theoretic in nature; we will see that the effectively defined maps on formulas are in fact induced by rather simple transformations on modal automata.

Modal logic provides a simple, yet surprisingly powerful, language for specifying properties of coalgebras. In this talk I introduce the basic modal logic that is interpreted on relational structures. My aim is to provide an idea how modal logic relates to other logics, such as first-order and intuitionistic logic, and to the duality between algebraic and coalgebraic structures.

If time permits, I might also give a very informal warm-up for the modal mu-calculus which is the topic of next week's talk.

The algebras of many-valued Lukasiewicz logics (MV algebras) as well as the theory of quantum measurement (Effect algebras) have undergone considerable development in the 1980s and 1990s; they now constitute important research fields, with connections to several contemporary areas of mathematics, logic, and theoretical computer science.

Both subjects have recently attracted considerable interest among groups of researchers in categorical logic and foundations of quantum computing. I will give a leisurely introduction to MV algebras (and their associated logics), as well as the more general world of effect algebras. If time permits, we will also illustrate some new results (with Mark Lawson, Heriot-Watt) on coordinatization of some concrete MV-algebras using inverse semigroup theory.

I shall give a brief introduction to System F.
I will then explain how to capture our intuition about polymorphic functions behaving uniformly by relational parametricity, and talk about ongoing work to find a notion of proof-relevant parametricity.

I will give a basic introduction to data types and initial-algebra semantics. The meaning of a data type is given as the initial object in a category of types with the corresponding constructors. Initiality immediately allows the modelling of a non-dependent recursion principle. I'll show how this can be upgraded to full dependent elimination, also known as induction, by using the uniqueness of the mediating arrow; in fact, induction is equivalent to recursion plus uniqueness. All possibly unfamiliar terms in this abstract will also be explained.

The core subject of Computer Science is "generated behaviour" (quiz: who said this?). Coalgebra provides the categorical formalisation of generated behaviour. I am planning to provide a first, very basic introduction to coalgebra. This will consist of two parts: i) coinduction & corecursion as means to define & reason about the (possibly) infinite behaviour of things; ii)modal logics for coalgebras.

The techniques used by the generic programming community have taught us that we can greatly benefit from exposing the common internal structure of a family of objects. One can for instance derive once and for all a wealth of iterators from an abstract characterisation of recursive datatypes as fixpoints of functors.

Our previous work on type and scope preserving semantics and their properties has made us realise that numerous semantics of the lambda calculus can be presented as instances of the fundamental lemma associated to an abstract notion of 'Model'. This made it possible to avoid code duplication as well as prove these semantics' properties generically.

Putting these two ideas together, we give an abstract description of syntaxes with binding making both their recursive and scoping structure explicit. The fundamental lemma associated to these syntaxes can be instantiated to provide the user with proofs that its language is stable under renaming and substitution as well as provide a way to easily define various evaluators.

For a category C we consider the endomorphism category End(C) and the subcategory of automorphisms Aut(C) -> End(C). It has been observed that for C the category of finite sets, finite dimensional vector spaces, or compact metric spaces this inclusion functor admits a simultaneous left and right adjoint.

We give general criteria for the existence of such adjunctions for a broad class of categories which includes FinSet, FinVect and CompMet as special cases. This is done using the language of factorisation systems and by introducing a notion of eventual image functors which provide a general method for constructing adjunctions of this kind.

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi have shown that, given a suitable distribution law, a pair of Hopf algebras forms two Frobenius algebras. Coming from the perspective of quantum theory, we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise \cite{Bonchi2014a} by including non-trivial dynamics of the underlying object – the so-called phase group – and investigate the effects of finite dimensionality of the underlying model, and recover the system of Bonchi et al as a subtheory in the prime power dimensional case. We show that the presence of a non-trivial phase group means that the theory cannot be formalised as a distributive law.

Theories featuring the interaction between a Frobenius algebra and a Hopf algebra have recently appeared in several areas in computer science: concurrent programming, control theory, and quantum computing, among others. Bonchi, Sobocinski, and Zanasi (2014) have shown that, given a suitable distributive law, a pair of Hopf algebras forms two Frobenius algebras. Here we take the opposite approach, and show that interacting Frobenius algebras form Hopf algebras. We generalise (BSZ 2014) by including non-trivial dynamics of the underlying object – the so-called phase group – and investigate the effects of finite dimensionality of the underlying model. We recover the system of Bonchi et al as a subtheory in the prime power dimensional case, but the more general theory does not arise from a distributive law.

I will sketch an alternative approach to economic game theory based upon the computer science idea of compositionality: concretely we i) give a number of operators for building up complex and irregular games from smaller and simpler games; and ii) show how the Nash equilibrium of these complex games can be defined recursively from their simpler components. We apply compositional reasoning to sophisticated games where agents must reason about how their actions affect future games and how those future games effect the utility they receive. This forces us into a second innovation — we augment the usual lexicon of games with a dual notion to utility because, in order for games to accept utility, this utility must be generated by other games. Our third innovation is to represent our games as string diagrams so as to give a clear visual interface to manipulate them. Our fourth, and final, innovation is a categorical formalisation of these intuitive diagrams which ensures our reasoning about them is fully rigorous.

The talk will be general so as appeal to as wide an audience as possible. In particular, no knowledge of category theory will be assumed!

I'll show how to generalise some results from algebra (think groups, rings, R-modules etc.) to a categorical setting using factorisation systems and an appropriate notion of finiteness on the objects of a category.

Formal languages and automata are taught to every computer science student. However, the student will most likely not see the beautiful coalgebraic foundations.

In this talk, I recapitulate how infinite trees can represent formal languages (sets of strings). I explain Agda's coinduction mechanism based on copatterns and demonstrate that it allows an elegant representation of the usual language constructions like union, concatenation, and Kleene star, with the help of Brzozowski derivatives. We will also investigate how to reason about equality of languages using bisimulation and coinductive proofs.

ThreadSafe is a static analysis tool for finding bugs in concurrent Java code that has been used by companies across the world to analyse and find bugs in large mission industrial applications. I will talk about how ThreadSafe works, and our experiences in applying static analysis technology to the "real world".

Motivated by the desire to understand the combinatorics of graph rewriting systems, it proved necessary to invent a formulation of graph rewriting itself that is not based on category theoretic structures as in the traditional framework, but on the concept of diagrammatic combinatorial Hopf algebras and reductions thereof. In this talk, I will present how the classical example of the Heisenberg-Weyl algebra of creation and annihilation of indistinguishable particles, which can alternatively be interpreted as the algebra of discrete graph rewriting, gave the initial clues for this novel framework. In hindsight, to pass from the special case of discrete graph rewriting to the case of general graph rewriting required every aspect of the framework of diagrammatic combinatorial Hopf algebras as a guideline for the construction, yet none of the traditional category theoretic ideas, whence one might indeed consider this reformulation as an independent formulation of graph rewriting.

The new framework results in a number of surprising results even directly from the formulation itself: besides the two main variants of graph rewriting known in the literature (DPO and SPO rewriting), there exist two more natural variants in the new framework. For all four variants, graph rewriting rules are encoded in so-called rule diagrams, with their composition captured in the form of diagrammatic compositions followed by one of four variants of reduction operations. Besides the general structure theory of the resulting algebras aka the rule algebras, one of the most important results to date of this framework in view of applications is the possibility to formulate stochastic graph rewriting systems in terms of the canonical representations of the rule algebras. Notably, this is closely analogous to the formulation of chemical reaction systems in terms of the canonical representation of the Heisenberg-Weyl algebra aka the bosonic Fock space. The presentation will not assume any prior knowledge of the audience on the particular mathematics required for this construction, and will be given on the whiteboard. The work presented is the result of a collaboration with Vincent Danos and Ilias Garnier (ENS Paris/LFCS University of Edinburgh), and (in an earlier phase) with Tobias Heindel (University of Copenhagen).

In this talk, we explore the fundamental category-theoretic structure needed to model relational parametricity (i.e., the fact that polymorphic programs preserve all relations) for the polymorphic lambda calculus (a.k.a. System F). Taken separately, the notions of categorical model of impredicative polymorphism and relational parametricity are well-known (lambda2-fibrations and reflexive graph categories, respectively). Perhaps surprisingly, simply combining these two structures results in a notion that only enjoys the expected properties in case the underlying category is well-pointed. This rules out many categories of interest (e.g. functor categories) in the semantics of programming languages.

To circumvent this restriction, we modify the definition of fibrational model of impredicative polymorphism by adding one further ingredient to the structure: comprehension in the sense of Lawvere. Our main result is that such comprehensive models, once further endowed with reflexive-graph-category structure, enjoy the expected consequences of parametricity. This is proved using a type-theoretic presentation of the category-theoretic structure, within which the desired consequences of parametricity are derived. Working in this type theory requires new techniques, since equality relations are not available, so that standard arguments that exploit equality need to be reworked.

This is joint work with Neil Ghani and Alex Simpson, and a dry run for a talk in Cambridge the week after.

I have recently begun to learn about the Cubical Type Theory of Coquand et al., as an effective computational basis for Voevodsky's Univalent Foundations, inspired by a model of type theory in cubical sets. It is in some ways compelling in its simplicity, but in other ways intimidating in its complexity. In order to get to grips with it, I have begun to develop my own much less subtle variation on the theme. If I am lucky, I shall get away with it. If I am unlucky, I shall have learned more about why Cubical Type Theory has to be as subtle as it is.

My design separates Coquand's all-powerful "compose" operator into smaller pieces, dedicated to more specific tasks, such as transitivity of paths. Each type path Q : S = T, induces a notion of value path s {Q} t, where either s : S, or s is •, "blob", and similarly, t : T or t = •. A "blob" at one end indicates that the value at that end of the path is not mandated by the type. This liberalisation in the formation of "equality" types allows us to specify the key computational use of paths between types, extrusion:

if Q : S = T and s : S, then s ⌢• Q : s {Q} •

That is, whenever we have a value s at one end of a type path Q : S = T, we can extrude that value across the type path, getting a value path which is s at the S end, but whose value at the T end is not specified in advance of explaining how to compute it. Extrusion gives us a notion of coercion-by-equality which is coherent by construction. It is defined by recursion on the structure of type paths. Univalence can be added to the system by allowing the formation of types interpolating two equivalent types, with extrusion demanding the existence of the corresponding interpolant values, computed on demand by means of the equivalence.

So far, there are disconcerting grounds for optimism, but the whole of the picture has not yet emerged: I may just have pushed the essential complexity into one corner, or the whole thing may be holed below the waterline. But if it does turn out to be nonsense, it will be nonsense for an interesting reason.

I will present work in progress on a (co)algebraic framework that allows to uniformly study dynamic modal logics such as Propositional Dynamic Logic (PDL) and Game Logic (GL). Underlying our framework is the basic observation that the program/game constructs of PDL/GL arise from monad structure, and that the axioms of these logics correspond to compatibility requirements between the modalities and this monad structure. So far we have a general soundness and completeness result for PDL-like logics wrt T-coalgebras for a monad T. I will discuss our completeness theorem, its limitations and plans for extending our results. [For the latter we might require the help of koalas, wallabies and wombats.]

Infinity-categories simultaneously generalise topological spaces and categories. Intuitively, a (weak) infinity-category should have objects, morphisms, 2-morphisms, 3-morphisms, ... and identity morphisms and composition which is suitably unital and associative up to a higher (invertible) morphism (the number 1 in (infinity, 1)-category means that k-morphisms for k > 1 are invertible) . The trouble begins when one naively tries to make these coherence conditions precise; already 4-categories famously requires 51 pages to define explicitly. Instead, one typically turns to certain "models" of infinity-categories that encode all this data implicitly, usually as some kind of simplicial object with additional properties. I will introduce two such models: quasicategories and complete Segal spaces. If time allows, I will also discuss hopes and dreams about internalising these notions in Type Theory, which should give a satisfactory treatment of category theory in Type Theory without assuming Uniqueness of Identity Proofs.

I've been working with Jules Hedges on a compositional model of game theory. After briefly reminding you of the model, I'll discuss where we are at – namely the definition of morphisms between games, and the treatment of choice and iteration of games. I'm hoping you will be able to shed some light on this murky area. There is a draft paper if anyone is interested.

String diagrams give a powerful graphical syntax for morphisms in symmetric monoidal categories (SMCs). They find many applications in computer science and are becoming increasingly relevant in other fields such as physics and control theory.

An important role in many such approaches is played by equational theories of diagrams, which can be oriented and used as rewrite systems. In this talk, I'll lay the foundations for this form of rewriting by interpreting diagrams combinatorially as typed hypergraphs and establish the precise correspondence between diagram rewriting modulo the laws of SMCs on the one hand and double pushout (DPO) rewriting of hypergraphs, subject to a soundness condition called convexity, on the other. This result rests on a more general characterisation theorem in which we show that typed hypergraph DPO rewriting amounts to diagram rewriting modulo the laws of SMCs with a chosen special Frobenius structure.

If there's time, I'll also discuss some of the results we have in developing the rewrite theory of hypergraphs for SMCs, namely termination proofs via graph metrics and strongly convex critical pair analysis.

In this talk, we consider extending Lawvere theories to allow enrichment in a base category such as CMonoid, Poset or Cat. In doing so, we see that we need to alter the formulation in a fundamental way, using the notion of cotensor, a kind of limit that is hidden in the usual accounts of ordinary category theory but is fundamental to enriched category theory. If time permits, we will briefly consider the specific issues that arise when one has two-dimensional structure in the enriching category, as exemplified by Poset and Cat.

In 1963, Bill Lawvere characterised universal algebra in category theoretic terms. His formulation being category theoretic was not its central contribution: more fundamental was its presentation independence. Three years later, monads were proposed as another category theoretic formulation of universal algebra. Overall, the former are technically better but the relationship is particularly fruitful and the latter are more prominent, cf Betamax vs VHS. So we study Lawvere theories carefully in the setting of universal algebra and in relation to monads.

The probabilistic calculus introduced in the eponymous paper by Robin Adams and Bart Jacobs is inspired by quantum theory by considering that conditional probabilities can be seen as side-effect-free measurements. A type-theoretic treatment of this semantic observation leads, once equipped with suitable computation rules, to the ability to do exact conditional inference.

I will present the type theory and the accompanying computation rules proposed in the paper and discuss some of the interesting open questions I will be working on in the near future.

2016-01-13: MSP101 Planning session (LT1310)

2015-12-02, 11:00: Introduction to coherence spaces, and how to use them for dependent session types (Bob Atkey, MSP)

Coherence spaces are a simplification of Scott domains, introduced by Girard to give a semantics to the polymorphic lambda-calculus. While investigating the structure of coherence spaces, Girard noticed that the denotation of the function type in coherence spaces can be decomposed into two independent constructions: a linear ("use-once") function space, and a many-uses-into-one-use modality. And so Linear Logic was discovered.

Coherence spaces are interesting because they model computation at a low level in terms of interchange of atomic 'tokens' of information. This makes them a useful tool for understanding several different computational phenomena.

In this talk, I'll show how coherence spaces naturally model session types, via Wadler's interpretation of Classical Linear Logic as a session-typed pi-calculus, and how that interpretation extends to an interpretation of a dependently typed version of session types.

I would like to speak about the categorical structure of the category of von Neumann algebras, with as morphisms normal, completely positive, unital linear maps. For some years my colleagues and I have worked on identifying basic structures in this category, and while surprisingly many things do not exist or do not work in this category (it's not a topos or even an extensive category, there's no epi-mono factorisation system, there is no dagger, colimits — if they exist at all — are horrendous...), we did find some structure (the products behave reasonable in some sense, there is a 'quotient', and 'comprehension', and we have a universal property for the minimal Stinespring dilation, and a universal property for M_2—the qubit). There is no deep category theory involved by any standards, and I promise I will spare you the functional analysis, so it should be a light talk.

Due to popular demand I am going to give a brief introduction to Marc Pauly's Coalition Logic, a propositional modal logic that allows to reason about the power of coalitions in strategic games. I will provide motivation and basic definitions. Furthermore I am planning to discuss how the logic can be naturally viewed as a coalgebraic logic and what we gain from the coalgebraic perspective. Finally — if (preparation) time permits — I am going to say how the logic can be applied to the area of mechanism design.

Commutative Frobenius algebras play an important role in both Topological Quantum Field Theory and Categorical Quantum Mechanics; in the first case they correspond to 2D TQFTs, while in the second they are non-degenerate observables. I will consider the case of "special" Frobenius algebras, and their associated group of phases. This gives rise to a free construction from the category of abelian groups to the PROP generated by this Frobenius algebra. Of course a theory with only one observable is not very interesting. I will consider how two such PROPs should be combined, and show that if the two algebras (i) jointly form a bialgebra and (ii) their units are "mutually real"; then they jointly form a Hopf algebra. This gives a "free" model of a pair of strongly complementary observables. I will also consider which unitary maps must exist in such models.

2015-10-22, 15:00 (LT1415): Semantics for Social Systems where Theory about the System Changes the System (Viktor Winschel, ETH Zurich)

In societies the notion of a law is not given by nature. Instead social dynamics are driven by the theories the citizens have about the dynamics of the social system. Obviously self-referential mathematical structures, developed in computer science, are candidates to be applied in social sciences for this foundational issue. We will see a prototypical game theoretical problem where several computer scientific tools can help to discuss these structures. It is a long standing problem in economics and of human kind and their scarce recourses: "should we go to a bar that is always so overcrowded"?

We introduce a notion of type and scope preserving semantics generalising Goguen and McKinna's "Candidates for Substitution" approach to defining one traversal generic enough to be instantiated to renaming first and then substitution. Its careful distinction of environment and model values as well as its variation on a structure typical of a Kripke semantics make it capable of expressing renaming and substitution but also various forms of Normalisation by Evaluation as well as, perhaps more surprisingly, monadic computations such as a printing function.

We then demonstrate that expressing these algorithms in a common framework yields immediate benefits: we can deploy some logical relations generically over these instances and obtain for instance the fusion lemmas for renaming, substitution and normalisation by evaluation as simple corollaries of the appropriate fundamental lemma. All of this work has been formalised in Agda.

Research in the proof theory of dynamic logics has recently gained momentum. However, features which are essential to these logics prevent standard proof-theoretic methodologies to apply straightforwardly. In this talk, I will discuss the main properties proof systems should enjoy in order to serve as suitable environments for an inferential theory of meaning (proof-theoretic semantics). Then, I'll identify the main challenges to the inferential semantics research agenda posed by the very features of dynamic logics which make them so appealing and useful to applications. Finally, I'll illustrate a methodology generating multi-type display calculi, which has been successful on interesting case studies (dynamic epistemic logic, propositional dynamic logic, monotone modal logic).

Escalation is the behavior of players who play forever in the same game. Such a situation is a typical field for application of coinduction which is the tool conceived for reasoning in infinite mathematical structures. In particular, we use coinduction to study formally the game called dollar auction, which is considered as the paradigm of escalation. Unlike what is admitted since 1971, we show that, provided one assumes that the other agent will always stop, bidding is rational, because it results in a subgame perfect equilibrium. We show that this is not the only rational strategy profile (the only subgame perfect equilibrium). Indeed if an agent stops and will stop at every step, whereas the other agent keeps bidding, we claim that he is rational as well because this corresponds to another subgame perfect equilibrium. In the infinite dollar auction game the behavior in which both agents stop at each step is not a Nash equilibrium, hence is not a subgame perfect equilibrium, hence is not rational. Fortunately, the notion of rationality based on coinduction fits with common sense and experience. Finally the possibility of a rational escalation in an arbitrary game can be expressed as a predicate on games and the rationality of escalation in the dollar auction game can be proved as a theorem which we have verified in the proof assistant COQ. In this talk we will recall the principles of infinite extensive games and use them to introduce coinduction and equilibria (Nash equilibrium, subgame perfect equilibrium). We will show how one can prove that the two strategy profiles presented above are equilibria and how this leads to a "rational" escalation in the dollar auction. We will show that escalation may even happen in much simpler game named 0,1-game.

Social Choice functions are procedures used to aggregate the preferences of individuals into a collective decision. We outline two recent abstract approaches to SCFs: a recent sheaf treatment of Arrow's Theorem by Abramsky and a modal logic studied by Ulle Endriss and myself. We show how to relate the categorical modelling of Social Choice problems to said work in Modal Logic. This insight prompts a number of research questions, from the relevance of sheaf-like condition to the modelling of properties of SCFs on varying electorates.

This is a sequel to my last 101 where I spoke about describing cellular automata as algebras of a comonad on Set. I'll describe how we can make sense of "generalised cellular automata" (probabilistic/non-deterministic/quantum, for example) as comonads on other categories via distributive laws of monads and comonads.

The Polymorphic Blame Calculus (PBC) of Ahmed et al. (2011) combines polymorphism, as in System F, with type dynamic and runtime casts, as in the Blame Calculus. The PBC is carefully designed to ensure relational parametricity, that is, to ensure that type abstractions do not reveal their abstracted types. The operational semantics of PBC uses runtime sealing and syntactic barriers to enforce parametricity. However, it is an open question as to whether these mechanisms actually guarantee parametricity for the PBC. Furthermore, there is some question regarding what parametricity means in the context of the PBC, as we have examples that are morally parametric but not technically so. This talk will review the design of the PBC with respect to ensuring parametricity and hopefully start a discussion regarding what parametricity should mean for the PBC.

We give a technique to construct a final coalgebra out of modal logic. An element of the final coalgebra is a set of modal formulas. The technique works for both the finite and the countable powerset functors. Starting with a corecursive algebra, we coinductively obtain a suitable subalgebra. We see - first with an example, and then in the general setting of modal logic on a dual adjunction - that modal theories form a corecursive algebra, so that this construction may be applied.

We generalize the framework to categories other than Set, and look at examples in Poset and in the opposite category of Set.

Relational parametricity is a fundamental concept within theoretical computer science and the foundations of programming languages, introduced by John Reynolds. His fundamental insight was that types can be interpreted not just as functors on the category of sets, but also as equality preserving functors on the category of relations. This gives rise to a model where polymorphic functions are uniform in a suitable sense; this can be used to establish e.g. representation independence, equivalences between programs, or deriving useful theorems about programs from their type alone.

The relations Reynolds considered were proof-irrelevant, which from a type theoretic perspective is a little limited. As a result, one might like to extend his work to deal with proof-relevant, i.e. set-valued relations. However naive attempts to do this fail: the fundamental property of equality preservation cannot be established. Our insight is that just as one uses parametricity to restrict definable elements of a type, one can use parametricity of proofs to ensure equality preservation. The idea of parametricity for proofs can be formalised using the idea of 2-dimensional logical relations. Interestingly, these 2-dimensional relations have clear higher dimensional analogues where (n+1)-relations are fibered over a n-cube of n-relations. Thus the story of proof relevant logical relations quickly expands into one of higher dimensional structures similar to the cubical sets which arises in Homotopy Type Theory. Of course, there are also connections to Bernardy and Moulin's work on internal parametricity.

There exists various possible methods to distribute seats proportionally between states (or parties) in a parliament. Hamilton's method (also known as the method of largest reminder) was abandoned in the USA because of some drawbacks, in particular the possibility of the Alabama paradox, but it is still in use in many other countries.

In recent work (joint with Svante Janson) we give, under certain assumptions, a closed formula for the probability that the Alabama paradox occurs given the vector p_1,...,p_m of relative sizes of the states.

From the theorem we deduce a number of consequences. For example it is shown that the expected number of states that will suffer from the Alabama paradox is asymptotically bounded above by 1/e. For random (uniformly distributed) relative sizes p_1,...,p_m the expected number of states to suffer from the Alabama paradox converges to slightly more than a third of this, or approximately 0.335/e=0.123, as m -> infinity.

I will assume no prior knowledge of electoral mathematics, but begin by giving a brief background to various methods suggested and used for the distribution of seats proportionally in a parliament (it's all in the rounding).

We define a category whose morphisms are 'games relative to a continuation', designed to allow games to be built recursively from simple components. The resulting category has interesting structure similar to (but weaker than) compact closed, and comes with an associated string diagram language.

In pure mathematics, cyclic homology is an invariant of associative algebras that is motivated by algebra, topology and even mathematicial physics. However, when studied from an abstract point of view it turns out to be an invariant of a pair of a monad and a comonad that are related by a mixed distributive law, and I speculate that this could lead to some potential applications in computer science.

Let's say you have a database of people's private information. For SCIENCE, or some other reason, you want to let third parties query your data to learn aggregate information about the people described in the database. However, you have a duty to the people whose information your database contains not to reveal any of their individual personal information. How do you determine which queries you will let third parties execute, and those you will not?

"Differential Privacy" defines a semantic condition on probabilistic queries that identifies queries that are safe to execute, up to some "privacy budget".

I'll present the definition of differential privacy, talk a bit about why it is better than some 'naive' alternatives (e.g., anonymisation), and also describe how the definition can be seen as an instance of relational parametricity.

A good place to read about the definition of differential privacy is the book "The Algorithmic Foundations of Differential Privacy" by Cynthia Dwork and Aaron Roth.

Classical computation, invertible computation, probabilistic computation, and quantum computation, form increasingly more sophisticated labelled transition systems. How can we approximate a transition system by less sophisticated ones? Considering all ways to get probabilistic information out of a quantum system leads to domain-theoretic ideas, that also apply in the accompanying Boolean logic. I will survey to what extent these domains characterise the system, leading with examples from quantum theory, in a way that is accessible to a broad audience of computer scientists, mathematicians, and logicians.

2015-03-11, 11:00: Collapsing (Peter Hancock)

The topic comes from theory of infinitary proofs, and cut-elimination. In essence it is about nicely-behaved maps from higher "infinities" to lower ones, as the infinitary proofs are er, infinite, and can be thought of as glorified transfinite iterators. What might nice behaviour mean?

You can think of it as how to fit an uncountable amount of beer into a bladder whose capacity is merely countable. (Or maybe even finite.)

The most ubiquitous form of infinity is the regular cardinal, iepassing from a container F to F + (mu F -> _), where mu is the W-type operation. I'll "explain" regular collapsing as being all about diagonalisation.

I gave an SPLS talk, which was mostly propaganda, about why people should stop claiming that totality loses Turing completeness. There was some technical stuff, too, about representing a recursive definition as a construction in the free monad whose effect is calling out to an oracle for recursive calls: that tells you what it is to be recursive without prejudicing how to run it. I'm trying to write this up double-quick as a paper for the miraculously rubbery MPC deadline, with more explicit attention to the monad morphisms involved. So I'd be grateful if you would slap down the shtick and make me more morphic. The punchline is that the Bove-Capretta domain predicate construction is a (relative) monad morphism from the free monad with a recursion oracle to the (relative) monad of Dybjer-Setzer Induction-Recursion codes. But it's worth looking at other monad morphisms (especially to the Delay monad) along the way.

What structure is required of a set so that computations in a given notion of computation can be run statefully this with set as the state space? Some answers: To be able to serve stateful computations, a set must carry the structure of a lens; for running interactive I/O computations statefully, a "responder-listener" structure is necessary etc. I will observe that, in general, to be a runner of computations for an algebraic theory (defined as a set equipped with a monad morphism between the corresponding monad and the state monad for this set) is the same as to be a comodel of this theory, ie a coalgebra of the corresponding comonad. I will work out a number of instances. I will also compare runners to handlers.

It would be a great shame if dependently-typed programming (DTP) restricted us to only writing very clever programs that were a priori structurally recursive and hence obviously terminating. Put another way, it is a lot to ask of the programmer to provide the program and its termination proof in one go, programmers should also be supported in working step-by-step. I will show a technique that lowers the barrier of entry, from showing termination to only showing productivity up front, and then later providing the opportunity to show termination (convergence). I will show an example of a normaliser for STLC represented in Agda as a potentially non-terminating but nonetheless productive corecursive function targeting the coinductive delay monad.

I've managed to prove a theorem that I've been chasing for a while. The trouble, of course, was stating it. I'll revisit the motivation for extending type systems with an analysis of not just what things are but where, when, whose, etc. The idea is that typed constructions occur in one of a preordered set of worlds, with scoping restricted so that information flows only "upwards" from one world to another. Worlds might correspond to "at run time" and "during typechecking", or to computation in distinct cores, or in different stages, etc. What does the dependent function space mean in this setting? For a long time, I thought that each world had its own universal quantifier, for abstracting over stuff from that world. Failure to question this presumption is what led to failure to state a theorem I could prove. By separating quantifiers from worlds, I have fixed the problem. I'll show how to prove the key fact: if I can build something in one world and then move it to another, then it will also be a valid construction once it has arrived at its destination.

Monoidal categories are essentially 2-dimensional things, so why on earth would we represent them using a linear string of symbols? I'll talk about how to use string diagrams for monoidal categories, graph rewriting for reasoning within them, and how the syntax can be extended to handle certain kinds of infinitary expressions with the infamous !-box. If there's time I'll finish with some half-baked (eh... basically still looking for the on switch of the oven...) ideas of how to generalise them.

The topic is the unholy trinity of eta, zeta, and xi. I'll indicate how Curry managed to give a finite combinatorial axiomatisation of this nastiness, by anticipating (almost-but-not-quite) McBride et al's applicative functors.

2014-11-06, 09:00: Comonadic Cellular Automata (Kevin Dunne, MSP)

Kevin will be giving an informal talk about some of the stuff he has been learning about. He'll give the definition of a cellular automaton and then talk about how this definition can be phrased in terms of a comonad.

Logical relations are widely used to study various properties of typed lambda calculi. By extending them to the lambda calculus with monadic types, we can gain understanding of the properties on functional programming languages with computational effects. Among various constructions of logical relations for monads, I will talk about a categorical TT-lifting, which is a semantic analogue of Lindley and Stark's leapfrog method.

After reviewing some fundamental properties of the categorical TT-lifting, we apply it to the problem of relating two monadic semantics of a call-by-value functional programming language with computational effects. This kind of problem has been considered in various forms: for example, the relationship between monadic style and continuation passing style representations of call-by-value programs was studied around '90s. We give a set of sufficient conditions to solve the problem of relating two monadic semantics affirmatively. These conditions are applicable to a wide range of such problems.

All kinds of semantics are syntax directed: the semantics follows from the syntax. Some varieties of semantics are syntax and type directed. In this talk, I'll discuss syntax, type, and analysis directed semantics (analysis-directed semantics for short!), for analyses other than types. An analysis-directed semantics maps from terms coupled with derivations of a static program analysis into some semantic domain. For example, the simply-typed lambda calculus with an effect system maps to the category generated by a strong parametric effect monad (due to Katsumata) and a bounded-linear logic-like analysis (described as a coeffect systems) maps to a category generated by various structures related to monoidal comonads. I'll describe a general technique for building analysis-directed semantics where semantic objects and analysis objects have the same structure and are coupled by lax homomorphisms between them. This aids proving semantic properties: the proof tree of an equality for two program analyses implies the rules needed to prove equality of the programs' denotations.

Neil will talk about partial progress made during the summer on higher dimensional parametricity and the cubical structures that seem to arise.
Details will be kept to a minimum and, of course, concepts stressed.

Or: My summer with Steve
Or: How Christine and Frank were right, after all
Or: Inductive types for the price of function extensionality and impredicative Set

Christine Paulin-Mohring and Frank Pfenning suggested to use impredicative encodings of inductive types in the Calculus of Constructions, but this was later abandoned, since it is "well-known" that induction principles, i.e. dependent elimination, can not be derived for this encoding. It now seems like it is possible to give a variation of this encoding for which the induction principle is derivable after all. The trick is to use identity types to cut down the transformations of type (Pi X : Set) . (F(X) -> X) -> X to the ones that are internally strongly dinatural, making use of a formula for a "generalised Yoneda Lemma" by Uustalu and Vene.

Ohad gave an informal overview of his current draft, with the following abstract:

Haskell incorporates computational effects modularly using sequences of monad transformers, termed monad stacks. The current practice is to find the appropriate stack for a given task using intractable brute force and heuristics. By restricting attention to algebraic stack combinations, we provide a linear-time algorithm for generating all the appropriate monad stacks, or decide no such stacks exist. Our approach is based on Hyland, Plotkin, and Power's algebraic analysis of monad transformers, who propose a graph-theoretical solution to this problem. We extend their analysis with a straightforward connection to the modular decomposition of a graph and to cographs, a.k.a. series-parallel graphs.

We present an accessible and self-contained account of this monad-stack generation problem, and, more generally, of the decomposition of a combined algebraic theory into sums and tensors, and its algorithmic solution. We provide a web-tool implementing this algorithm intended for semantic investigations of effect combinations and for monad stack generation.

This 101 is intended to be a brainstorming session on possible links between the theory of coalgebras and the theory of databases. I will outline some ideas in this direction and I am looking forward to your feedback.

In Intuitionistic Multiplicative Linear Logic, the right introduction rule for tensors implies picking a 2-partition of the set of assumptions and use each component to inhabit the corresponding tensor's subformulas. This makes a naive proof search algorithm intractable. Building a notion of resource availability in the context and massaging the calculus into a more general one handling both resource consumption and a notion of "leftovers" of a subproof allows for a well-structured well-typed by construction proof search mechanism.

Guillaume presented parts of Hedges' paper Monad transformers for backtracking search (accepted to MSFP 2014). The paper extends Escardo and Oliva's work on the selection and continuation monads to the corresponding monad transformers, with applications to backtracking search and game theory.

Neil spoke about how adding structured quotients to containers gives rise to a larger class of data types.

2014-02-19, 11:00: Synthetic Differential Geometry (Tim Revell, MSP)

Tim gave a brief introduction to Synthetic Differential Geometry. This is an attempt to treat smooth spaces categorically so we can extend the categorical methods used in the discrete world of computer science to the continuous work of physics.

In 1987, Felleisen showed how to add control operators (for things like exceptions and unconditional jumps) to the untyped lambda-calculus. In 1990, Griffin idly wondered what would happen if one did the same in a typed lambda calculus. The answer came out: the inhabited types become the theorems of classical logic.

I will present the lambda mu-calculus, one of the cleanest attempts to add control operators to a type theory. We'll cover the good news: the inhabited types are the tautologies of minimal classical logic, and Godel's Double Negation translation from classical to intuitionistic logic turns into the CPS translation.

And the bad news: control operators don't play well with other types. Add natural numbers (or some other inductive type), and you get inconsistency. Add Sigma-types, and you get degeneracy (any two objects of the same type are definitionally equal). It gets worse: add plus-types, and you break Subject Reduction.